Explore the vast range of titles available.

Fermatean fuzzy sets (FFSs) have become a potent tool for modeling uncertainty in decision-making, offering a higher degree of flexibility compared to traditional fuzzy and intuitionistic fuzzy sets (IFSs). Within the context of evidence theory, the notions of belief and plausibility enhance the representational capacity of FFSs, enabling the handling of ambiguous and conflicting data. This study proposes advanced constructions of distance and similarity measures for belief and plausible Fermatean fuzzy sets (BP-FFSs). These measures are designed to capture nuanced differences and relationships among BP-FFSs, addressing critical gaps in existing methodologies. The proposed measures are rigorously evaluated for essential mathematical properties. Furthermore, their effectiveness is validated through real-world applications in fields such as pattern identifications, multi-criteria decision-making (MCDM) and medical diagnosis. Numerical analyses using variety of problems demonstrate the superiority and robustness of the suggested approach in handling complex uncertainty scenarios. This research contributes to the theoretical foundation of fuzzy set theory (FST) and its practical applications, offering a comprehensive framework that advances both the understanding and utilization of BP-FFSs. The findings underscore the potential of the proposed measures in addressing contemporary challenges across diverse domains.

Model selection in data science involves evaluating multiple alternatives across conflicting criteria under uncertainty, where existing fuzzy multi criteria group decision making (MCGDM) approaches often fail to capture asymmetric uncertainty and nonlinear interactions in expert judgments. To address this limitation, this study proposes a novel MCGDM framework based on fractional orthopair fuzzy sets (FOFS). The FOFS structure enables flexible and precise modeling of uncertainty by allowing independent fractional control of membership degree (MD) and non-membership degree (NMD). Furthermore, sine trigonometric aggregation operators are introduced to capture nonlinear relationships and fluctuations in expert evaluations. An integrated FOFS–TOPSIS method is then developed to rank candidate models based on their distances from positive ideal solution (PIS) and negative ideal solution (NIS). The applicability of the framework is demonstrated through a numerical study involving fifteen predictive models, fifteen evaluation criteria, and three experts. The results indicate that Alternative achieved the highest overall ranking, followed by , while mid and lower ranked models revealed trade-offs in accuracy, computational efficiency, and robustness. Comparative and sensitivity analyses confirm the framework’s robustness, stability, and superior ranking performance.

Non-Newtonian materials like the Jeffery fluid model (JFM) are crucial in many industries, such as food and fiber optics. In this analysis, a mathematical framework is proposed to describe the performance of a two-dimensional magnetohydrodynamic Jeffery flow model (MHD-JFM) across an exponential and contracting sheet under the influence of destructive and generative chemical reactions. To facilitate this analysis, a couple of difference equations is transformed into ordinary equations by applying the resemblance alteration. To obtain a numerical solution for the various fields of interest, namely velocity components and mass transfer at the surface (MTS), the bvp4c method is employed. The ground-breaking aspect of this study lies in its meticulous examination of the influence and stability of the Lorentz force on Jeffery fluid, particularly in the context of extending or contracting sheets that experience internal heat transfer as well as destructive and generative chemical reactions. Remarkably, such a comprehensive investigation has not yet been undertaken in the existing literature. Consequently, the present work is validated by comparing with available work. A stability investigation is conducted to ensure the reliability of the first solution. Through the utilization of graphs, the impact of factors like the Schmidt number, Hartmann number, destructive and generative chemical reaction factor, and Deborah numbers on the velocity component and mass transfer is thoroughly examined and discussed. It is investigated that mass transfer surfaces are a diminishing function of Deborah numbers. It has been detected that as the magnitude of escalates from 0.2 to 0.8, the velocity contours exhibit a declining trend across the boundary layer. A decreasing behaviour is observed for the velocity profiles as the quantities of Hartmann number enhances from 0.2 to 0.6. It can be observed that mass transfer exhibits a diminishing trend in response to destructive chemical reactions, whereas mass transfer experiences an increasing trend for generative chemical reactions.

Sustainable energy planning is a critical challenge, particularly in regions with complex decision-making environments and uncertain data. The selection of an optimal energy source requires robust methodologies that can effectively handle multi-criteria decision-making (MCDM) under uncertainty. This study explores the application of the multiple objective optimization on the basis of ratio analysis plus full multiplicative form (MULTIMOORA) method by incorporating the concept of the complex linear Diophantine fuzzy ( ) set. The set extends the conventional linear Diophantine fuzzy set by introducing a phase component, thereby enhancing the system’s adaptability. To examine the interrelationships among multiple numbers, we develop the Dombi prioritized averaging ( ) and the Dombi prioritized geometric ( ) aggregation operators, along with their weighted versions, based on the proposed Dombi operational laws. The fundamental properties of these aggregation operators are systematically analyzed. The developed operators are then integrated into the MULTIMOORA method to address MCDM problems. To illustrate the practical effectiveness of the proposed framework, a case study is conducted to determine a sustainable energy source for Gwadar, Pakistan, utilizing information. Furthermore, comparative analyses are performed against existing methodologies to validate the applicability and accuracy of the proposed approach.

Sustainable urban water management is increasingly challenged by uncertainty, imprecision, and hesitancy in evaluating alternative water sources. This study proposes a novel multi-criteria decision-making (MCDM) framework based on fractional orthopair fuzzy (FOF) sets, designed to model partial hesitancy and fractional expert judgments more effectively than traditional fuzzy methods. Integrating an entropy-based weighting scheme and the technique for order preference by similarity to ideal solution (TOPSIS), the framework is applied to assess water resource alternatives in Lahore, Pakistan a city facing rapid groundwater depletion, urban expansion, and declining surface water quality. The evaluation considers three key criteria: water quality, availability, and affordability across the alternatives of surface water, groundwater, and rainwater. Results show that rainwater harvesting is the most sustainable option, with a closeness coefficient of , outperforming alternatives in terms of both cost-effectiveness and safety. Sensitivity analysis on parameters ( , ) confirms the model’s robustness. The findings offer actionable guidance for water authorities, emphasizing the importance of rainwater harvesting and reduced reliance on depleting groundwater. The proposed model is adaptable to other urban regions, provided expert input and contextual data are available.

In this paper, a new collocation method based on Haar wavelet is developed for numerical solution of Riccati type differential equations with non-integer order. The fractional derivatives are considered in the Caputo sense. The method is applied to one test problem. The maximum absolute estimated error functions are calculated, and the performance of the process is demonstrated by calculating the maximum absolute estimated error functions for a distinct number of nodal points. The results show that the method is applicable and efficient.

An analytical method for solving the fractional diffusion–wave equation with a reaction is investigated. This approach is based on the Laplace transform and fractional series method. An analytical derivation for the proposed method is presented. Examples are given to illustrate the efficiency of the method. The obtained solutions are very close to the exact solutions. Based on this study, we think that the obtained method is promising, and we hope that it can be implemented to other physical problems.

We study the solution of fractional Fredholm integrodifferential equation. A modified version of the fractional power series method (RPS) is presented to extract an approximate solution of the model. The RPS method is a combination of the generalized fractional Taylor series and the residual functions. To show the efficiency of the proposed method, numerical results are presented.

In this paper, we investigate a numerical solution of Lienard’s equation. The residual power series (RPS) method is implemented to find an approximate solution to this problem. The proposed method is a combination of the fractional Taylor series and the residual functions. Numerical and theoretical results are presented.

In this article, the solution of the time-fractional Phi-4 equation is investigated. We implement the residual power series method to approximate the solution of this equation. Numerical results are presented. In addition, the effects of the fractional order on the Phi-4 are discussed graphically.